Third degree price discrimination in linear-demand markets: effects on number of markets served and social welfare

Southern Economic Journal, Oct, 2008 by Victor Kaftal, Debashis Pal

To illustrate, let us revisit the case of three markets presented in Proposition 3. As already mentioned, price discrimination always lowers welfare if all markets are served (k = 1); that is, [V.sub.1] [greater than or equal to] [V.sub.2] and [V.sub.1] [greater than or equal to] [V.sub.3], and always increases it if only one market is served (k = 3); that is, [V.sub.3] > [V.sub.1] and [V.sub.3] > [V.sub.2]. If only markets 2 and 3 are served, then Proposition 5 states that price discrimination increases welfare if and only if [S.sub.1] > (1/3)([S.sub.2] [S.sub.3] - [V.sub.2]); that is, if and only if

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The inequalities characterizing the cases by Proposition 3 are best illustrated by Figures 3 and 4, which illustrate the areas where welfare increases with price discrimination in terms of the variables [S.sub.1] and [[alpha].sub.1].

Notice that if the size [S.sub.1] is large (or if not, if [[alpha].sub.1] is sufficiently large to compensate), market 1 will be served (see Proposition 4). But in case it is not, price discrimination will benefit welfare only if [S.sub.1] (or [[alpha].sub.1]) are not too small.

The policy makers often have less information about the market demands than the monopolist. Among the available information is the number of markets currently served by the monopolist and a reasonable estimate of the market sizes. This information alone may be sufficient to evaluate the impact of price discrimination as shown by the following example.

[FIGURE 3 OMITTED]

EXAMPLE 1. For n = 3, if only markets 2 and 3 are served, then welfare increases with price discrimination if [S.sub.1] > (1/3)min([S.sub.2], [S.sub.3]).

[FIGURE 4 OMITTED]

The sufficient condition in Example 1 generalizes to any number of markets.

COROLLARY l. Suppose at least j markets are not served under uniform pricing. If j = n - 1, price discrimination always increases welfare, and if j < n - 1, price discrimination increases welfare if

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5. Continuous Distribution of Linear Markets

There are two reasons to consider a continuous distribution of markets instead of a finite number of markets as we have done so far. The first one is that continuous markets may arise naturally. For instance, as Cowan (2007) has pointed out, Coca Cola considered a marketing plan (never implemented) of pricing its vending machine products according to the local temperature--a continuous parameter. The second one is that a continuous distribution provides a simpler and, in the following sense, a less arbitrary way to represent a large number of markets. Indeed, as we have pointed out in section 2, if two markets have the same [alpha], they should be combined in our model into a single market. It is, therefore, arbitrary to distinguish markets with small differences between their as. Rather, it is more intuitive and more appropriate to combine all markets whose [alpha]s fall within the same (small) intervals. In this familiar Riemann sum limiting process, we can describe the system of markets via a size distribution function S([alpha]), that is, [[integral].sup.b.sub.a] S([alpha]) d[alpha] represents the total size of the markets with a [less than or equal to] [alpha] [less than or equal to] b. This is equivalent to considering a continuum of linear markets with size a piecewise continuous function S([alpha]) [greater than or equal to] 0, for 0 [less than or equal to] [alpha] [less than or equal to] A. For example, S([alpha]) = (1/2)[alpha] for [alpha] [member of] [1 - x, 1 x] and S ([alpha]) = 0 otherwise, give rise to the rotating demands considered in Malueg and Schwartz (1994). Then all the results in sections 2, 3, and 4 and in the Appendix will go through essentially unchanged if we substitute sums with integrals. So the aggregate demand, profit, and welfare functions are for all 0 [less than or equal to] p [less than or equal to] A: